Normalized Solutions for the Kirchhoff-Schrödinger Systems with Weakly Attractive Potentials

In this paper, we consider the following Kirchhoff-Schrödinger system with weakly attractive potentials:

{

−(

a+b

∫

ℝ

N

|

∇

u

1

|

2

dx

)Δ

u

1

+

V

1

u

1

=

λ

1

u

1

+

ν

1

|

u

1

|

p

1

−2

u

1

+α

q

1

|

u

1

|

q

1

−2

u

1

|

u

2

|

q

2

,

−(

a+b

∫

ℝ

N

|

∇

u

2

|

2

dx

)Δ

u

2

+

V

2

u

2

=

λ

2

u

2

+

ν

2

|

u

2

|

p

2

−2

u

2

+α

q

2

|

u

1

|

q

1

|

u

2

|

q

2

−2

u

2

,

having prescribed mass

∫

ℝ

N

|

u

i

|

2

dx

=

m

i

, where

a>0

,

b,α,

ν

i

>0

,

q

i

>1

,

N=2,3

,

λ

i

∈ℝ

are Lagrange multiplier and

V

i

∈

C

1

(

ℝ

N

)

are potential functions for

i=1,2

. When

2+

8

N

<

q

1

+

q

2

<

2

*

and

(

p

1

,

p

2

)∈

ℝ

2

, we prove the existence of multiple solutions, which are positive radial vectors.

2

*

=

2N/

(

N−2

)

is the Sobolev critical exponent. The proof is based on variational techniques and constrained minimization arguments.

• Schrödinger Systems
• Normalized Solutions
• Weakly Attractive Potentials
• Variational Method
• Pohoaev Manifold

1. Introduction and Main Results

The focus of this paper is on the nonlocal Schrödinger system with weakly attractive potentials in H 1 ( ℝ N ) × H 1 ( ℝ N ) :

(1.1) { − ( a + b ∫ ℝ N | ∇ u 1 | 2 d x ) Δ u 1 + V 1 u 1 = λ 1 u 1 + ν 1 | u 1 | p 1 − 2 u 1 + α q 1 | u 1 | q 1 − 2 u 1 | u 2 | q 2 , − ( a + b ∫ ℝ N | ∇ u 2 | 2 d x ) Δ u 2 + V 2 u 2 = λ 2 u 2 + ν 2 | u 2 | p 2 − 2 u 2 + α q 2 | u 1 | q 1 | u 2 | q 2 − 2 u 2 ,

with the normalization constraint

(1.2) ∫ ℝ N | u 1 | 2 d x = m 1 > 0     and     ∫ ℝ N | u 2 | 2 d x = m 2 > 0 ,

where a , b , α , ν i > 0 for i = 1 , 2 . Assume q 1 , q 2 > 1 and 2 < p 1 , p 2 , q 1 + q 2 < 2 * . λ 1 , λ 2 ∈ ℝ are Lagrange multipliers. V 1 , V 2 ∈ C 1 ( ℝ N ) are potential functions satisfying

(V₁) lim | x | → ∞ V i ( x ) = sup x ∈ ℝ N V i ( x ) = 0 ≥ V i ( x ) and there exists ω i ∈ [ 0 , 1 / 2 ) such that | V i | N / 2 ≤ ω i S , where

S = inf u ∈ D 1 , 2 ( ℝ N ) \ { 0 } ∫ ℝ N | ∇ u | 2 d x ( ∫ ℝ N | u | 2 * d x ) 2 / 2 * ;

(V₂) set W i ( x ) : = ( ∇ V i ( x ) ⋅ x ) / 2 , W i ∈ C 1 ( ℝ N ) , lim | x | → ∞ W i ( x ) = 0 and there exists ρ i ∈ [ 0 , 1 ) such that | W i | N / 2 ≤ ρ i S .

(V₃) set Y i ( x ) : = γ q i q i W i ( x ) + Z i ( x ) , where Z i ( x ) : = ∇ W i ( x ) ⋅ x and Z i ∈ L s ( ℝ N ) for some s ∈ [ N / 2 , ∞ ] , there exists σ i such that | Y i , + | N / 2 ≤ σ i S , where Y i , + = max { Y i , 0 } .

An example satisfying the conditions (V₁)-(V₃) is V i ( x ) = − c | x | d + 1 , x ∈ ℝ N with constant d > 2 and suitable small constant c . Obviously, V = 0 also satisfies the conditions (V₁) (V₃).

For the study of Kirchhoff, Li, Luo and Yang [1] studied a Kirchhoff equation with a combined nonlinearity given by:

(1.3) { − ( a + b ∫ ℝ N | ∇ u | 2 d x ) Δ u + λ u = | u | p − 2 u + ν | u | q − 2 u ,     x ∈ ℝ N , ∫ ℝ N | u | 2 d x = c ,

where a , b , c , ν > 0 . They demonstrated the existence of multiple solutions when 2 < q < 10 3 and 14 3 < p < 6 , as well as ground states for the cases 2 < q < 10 3 < p = 6 and 14 3 < q < p ≤ 6 . Additionally, their work provided insights into the asymptotic behavior of the obtained solutions. In contrast, when ν ≤ 0 , Carrião, Miyagaki and Vicente [2] investigated the scenario. They established the existence of ground states for the equation when 2 < q < 2 * = p or 2 < q ≤ p ¯ = 14 3 < p < 2 * , offering complementary results to those of Li et al.

Ye [3] studied the nonautonomous Kirchhoff-type problem:

(1.4) { − ( a + b ∫ ℝ 3 | ∇ u | 2 d x ) Δ u + λ u = | u | p − 2 u + V ( x ) | u | q − 2 u ,     x ∈ ℝ N , ∫ ℝ N | u | 2 d x = c ,

where p = 14 3 , q = 4 , and V ∈ L loc ∞ ( ℝ 3 ) satisfies

V ( x ) ≥ 0 ,     lim | x | → ∞ V ( x ) = 0.

Using the concentration compactness principle, Ye demonstrated the existence of a minimizer under certain conditions. Specifically, the author established that there exist constants a 0 , b 0 , c 0 > 0 such that for a < a 0 , b < b 0 , and c < c 0 , the problem (1.4) admits a minimizer. This result highlights the role of the compactness argument in proving the existence of solutions in the presence of nonautonomous potentials.

Over the past two decades, systems, which similar to (1.1), have garnered significant attention due to their important physical background. When b > 0 , the ( b ∫ ℝ N | ∇ u | 2 d x ) Δ u , leading to its classification as a Kirchhoff-Schrödinger system, is nonlocal term of system (1.1). This nonlocal nature is associated with the following equation

τ ∂ 2 u ∂ t 2 − ( P 0 h + E 2 L ∫ 0 L | ∂ u ∂ x | 2 d x ) ∂ 2 u ∂ x 2 = 0

which was proposed by Kirchhoff in 1883 [4]. This equation not only serves as an extension of the classical D’Alembert’s wave equation but also relevant in biological contexts. Over the last few decades, Kirchhoff-type problems have drawn considerable attention, beginning with Lions’ foundational work [5], which established abstract framework for such problems. Since then, numerous important results have been developed, evidenced by works in [6]-[10]. For a more detailed discussion on the physical implications of systems such as (1.1), as well as additional physical and mathematical interpretations, we refer the reader to [11]-[13], along with the references cited in these works.

Building on this foundation, there has been growing interest in exploring the existence of normalized solutions. For example, the existence and multiplicity of normalized solutions for Schrödinger systems have garnered significant attention in recent years, as explored in studies such as [14]-[16] and related references. A notable example is the system

(1.5) { − Δ u + λ 1 u = ν 1 | u | p − 2 u + μ 1 | u | p 1 − 2 u + α q 1 | u | q 1 − 2 u | v | q 2 in   ℝ N , − Δ v + λ 2 v = ν 2 | v | p − 2 v + μ 2 | v | p 2 − 2 v + α q 2 | u | q 1 | v | q 2 − 2 v in   ℝ N ,

which has been analyzed under various parameter settings. For N ≥ 3 , when μ 1 = μ 2 = 0 , p = 4 , and q 1 = q 2 = 2 , the existence and multiplicity of normalized solutions were established in [17]-[19]. For N = 3 , 4 , when ν 1 = ν 2 = 0 , q 1 , q 2 > 1 , p 1 , q 1 + q 2 ∈ ( 2 , 2 * ) and p 2 ∈ ( 2 , 2 * ] , Li and Zou [20] investigated the associated Pohoaev manifold and proved the existence of a normalized solution. When N = 4 , p = 3 , p 1 , p 2 ∈ ( 2 , 4 ) and q 1 = q 2 = 2 , Luo et al. [21] analyzed the existence, nonexistence, and asymptotic properties of normalized solutions, particularly in the Sobolev critical case. More recently, Liu and Fang [15] studied the system for N = 3 , 4 , q 1 , q 2 > 1 and p 1 , p 2 , q 1 + q 2 ∈ ( 2 + 4 / N , 2 * ] , proving both the existence and nonexistence of normalized solutions. These studies illustrate the breadth of research on Schrödinger systems, addressing various nonlinearities, couplings, dimensions, and parameter regimes.

Furthermore, Hu and Mao [22] considered the following Kirchhoff-Schrödinger system

(1.6) { − ( a + b ∫ ℝ N | ∇ v 1 | 2 d x ) Δ v 1 = λ 1 v 1 + μ 1 | v 1 | p 1 − 2 v 1 + α q 1 | v 1 | q 1 − 2 v 1 | u 2 | q 2 , − ( a + b ∫ ℝ N | ∇ v 2 | 2 d x ) Δ v 2 = λ 2 v 2 + μ 2 | u 2 | p 2 − 2 v 2 + α q 2 | v 1 | q 1 | v 2 | q 2 − 2 v 2 ,

having prescribed mass ∫ ℝ N | v i | 2 d x = c i > 0 , where q i > 1 , a , b , α , μ i > 0 for i = 1 , 2 , 2 < p 1 , p 2 , q 1 + q 2 < 2 * and N = 2 , 3 . They established multiple positive radial vector solutions using variational methods combined with a constrained minimization approach.

Building on the results discussed earlier, a natural question emerges: does system (1.1) admit multiple normalized solutions? In this paper, we confirm this with a positive answer. Notably, when V ≠ 0 , the existing literature offers limited insights into system (1.1). Our objective is to address this gap by extending the findings of [22] to the Kirchhoff-Schrödinger system with potentials, thereby broadening the understanding of such systems.

Recently, growing attention has been directed toward the Kirchhoff-Schrödinger system, which is exemplified by (1.1). Similar to the case of the equation, where λ is considered an unknown Lagrange multiplier, the problem (1.1) can be viewed as a characteristic value issue. Within this framework, solving (1.1)-(1.2) involves analyzing constrained variational problems. Normalized solutions are then obtained by identifying the critical points of the energy functional I : H 1 ( ℝ N ) × H 1 ( ℝ N ) → ℝ , defined by

(1.7) I ( u 1 , u 2 ) = a 2 ∑ i = 1 2 | ∇ u i | 2 2 + b 4 ∑ i = 1 2 | ∇ u i | 2 4 + 1 2 ∑ i = 1 2   ∫ ℝ N V i u i 2 d x     − ∑ i = 1 2 ν i p i ∫ ℝ N | u i | p i d x − α ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x

on S ( m 1 ) × S ( m 2 ) , where S ( m ) : = { u ∈ H 1 ( ℝ N ) : | u | 2 2 = m } for all positive constant m .

To address compactness issues, we restrict our analysis to a radial framework. Specifically, we seek critical points of the functional I restricted on S r ( m 1 ) × S r ( m 2 ) , where

S r ( m ) : = { u ∈ H rad 1 ( ℝ N ) : ∫ ℝ N u 2 = m , ∫ ℝ N V u 2 < + ∞ }

and H rad 1 ( ℝ N ) represents the space of radial H 1 -functions on ℝ N . The radial restriction allows us to invoke compact embedding theorems, which are essential for the variational arguments. By invoking the principle of symmetric criticality, we ensure that critical points of I constrained to S r ( m 1 ) × S r ( m 2 ) are also critical points of I when constrained on the broader set S ( m 1 ) × S ( m 2 ) . Set

γ p : = N ( p − 2 ) 2 p .

Clearly, critical points of I | S r ( m 1 ) × S r ( m 2 ) lie within the Pohoaev manifold

ℳ : = { ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) : J ( u 1 , u 2 ) = 0 } ,

where

J ( u 1 , u 2 ) : = a ∑ i = 1 2 | ∇ u i | 2 2 + b ∑ i = 1 2 | ∇ u i | 2 4 − 1 2 ∑ i = 1 2   ∫ ℝ N W i u i 2 d x − ∑ i = 1 2   ν i γ p i | u i | p i p i       − α γ q 1 + q 2 ( q 1 + q 2 ) ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ,

which is the Pohoaev identity.

To present our results, we define the set ℬ ( τ ) : = { ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) : | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 < τ } for all τ > 0 and assume the following condition:

(M₁) 2 < p 1 , p 2 < 2 + 4 N , 2 + 8 N < q 1 + q 2 < 2 * .

The mathematical role of these constraints, such as ensuring the nonlinearities are well-behaved and avoiding compactness loss.

Now, the main result can be stated as follows.

Theorem 1.1.Given that condition(M₁)is satisfied. Subsequently, there exist three positive constants a * : = max { ( 2 max { ρ 1 , ρ 2 } + max { σ 1 , σ 2 } 2 − max { γ p 1 p 1 , γ p 2 p 2 } ) + max { ρ 1 , ρ 2 } ,                                 max { ρ 1 , ρ 2 } ( ∑ i = 1 2   ν i + α q ) ∑ i = 1 2   ν i ( 1 − γ p i ) , max { ω 1 , ω 2 } } , τ 0 = τ 0 ( m 1 , m 2 ) > 0 , α 0 = α 0 ( m 1 , m 2 ) ensuring that, for α ∈ ( 0 , α 0 ) and a > a * ,

(i) System(1.1)-(1.2)admits a solution ( u 1 , u 2 ) , which is positive radial vector, for some λ 1 , λ 2 < 0 . Additionally, I ( u 1 , u 2 ) > 0 ;

(ii) System (1.1)-(1.2) admits a solution ( v 1 , v 2 ) , which is positive radial vector, for some λ 1 , λ 2 < 0 . Additionally, ( v 1 , v 2 ) ∈ ℬ ( τ 0 ) and I ( v 1 , v 2 ) < 0 .

We now proceed to present the proof of Theorem 1.1. To prove Theorem1.1 (i), we begin by demonstrating that under the condition (M₁), the following inequality holds:

(1.8) inf ℬ ( τ ) I ( u 1 , u 2 ) < 0     for   any   τ > 0.

Additionally, we show the existence of constants τ 0 = τ 0 ( m 1 , m 2 ) > 0 , α 0 = α 0 ( m 1 , m 2 ) > 0 such that for any 0 < α ≤ α 0 , we have

(1.9) inf ∂ ℬ ( τ 0 ) I ( u 1 , u 2 ) > 0.

With these results in hand, a min-max structure of the mountain pass type can be introduced. Specifically, there exists τ ¯ ∈ ( 0 , τ 0 ) , ensuing that for all 0 < α ≤ α 0 , the desired conclusions follow that

(1.10) γ ( m 1 , m 2 ) : = inf η ∈ Γ max t ∈ [ 0 , 1 ] E ( η ( t ) ) > max { E ( η ( 0 ) ) , E ( η ( 1 ) ) } ,

where Γ : = { η ∈ ( S r ( m 1 ) × S r ( m 2 ) , C [ 0 , 1 ] ) : η ( 0 ) ∈ ℬ ( τ ¯ ) , η ( 1 ) ∉ ℬ ( τ 0 ) ¯ , E ( η ( 1 ) < 0 ) } . This approach allows us to search for a mountain pass solution. There are two primary challenges in the proof. One is to show the boundedness of ( P S ) γ ( m 1 , m 2 ) sequence { ( u 1 n , u 2 n ) } in S r ( m 1 ) × S r ( m 2 ) . By employing the method described in [23], we get a ( P S ) γ ( m 1 , m 2 ) sequence, along with the property that J ( u 1 n , u 2 n ) → 0 . Utilizing this property, we can show that the functional is coercive, leading to the ( u 1 n , u 2 n ) converges weakly to ( u 1 , u 2 ) in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) . The other is proving the compactness of ( P S ) γ ( m 1 , m 2 ) sequence. Establishing ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) is the crucial step. To address this requirement, we must excluding both the semi-trivial solutions and the trivial solutions of the problem (1.1), which introduces a complication not present in the case of Kirchhoff equation. In order to tackle this challenge, we employ the technique outlined in [[24], Lemma A.2] and integrating the uniqueness of positive solutions to Equation (2.1) with energy estimations.

In order to prove Theorem (1.1) (ii), we naturally introduce the following minimization problem for all 0 < α ≤ α 0 , derived from Equation (1.8) and Equation (1.9),

(1.11) c ( m 1 , m 2 ) : = inf ℬ ( τ 0 ) I ( u 1 , u 2 ) < 0.

Moreover, we define

(1.12) ϒ ( u 1 , u 2 ) ( θ ) : = I ( θ ∗ u 1 , θ ∗ u 2 ) = a 2 e 2 θ ∑ i = 1 2 | ∇ u i | 2 2 + b 4 e 4 θ ∑ i = 1 2 | ∇ u i | 2 4 + 1 2 ∑ i = 1 2   ∫ ℝ N V i ( e − θ x ) u i 2 d x     − ∑ i = 1 2 ν i p i e γ p i p i θ | u i | p i p i − α e γ q q θ ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ,

where θ ∗ u i : = e N 2 θ u i ( e θ x ) for i = 1 , 2 , q = q 1 + q 2 . It follows from ( ϒ ( u 1 , u 2 ) ) ′ ( 0 ) = J ( u 1 , u 2 ) that we divide ℳ into

ℳ + : = { ( u 1 , u 2 ) ∈ ℳ : ( ϒ ( u 1 , u 2 ) ) ( 0 ) > 0 } , ℳ 0 : = { ( u 1 , u 2 ) ∈ ℳ : ( ϒ ( u 1 , u 2 ) ) ( 0 ) = 0 } , ℳ − : = { ( u 1 , u 2 ) ∈ ℳ : ( ϒ ( u 1 , u 2 ) ) ( 0 ) < 0 } ,

where

( ϒ ( u 1 , u 2 ) ) ( 0 ) = 2 a ∑ i = 1 2 | ∇ u i | 2 2 + 4 b ∑ i = 1 2 | ∇ u i | 2 4 + 1 2 ∑ i = 1 2   ∫ ℝ N Z i u i 2 d x     − ∑ i = 1 2   ν i γ p i 2 p i | u i | p i p i − α ( γ q q ) 2 ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x .

In this paper, we define the L p -norm as | u | p : = ( ∫ ℝ N | u | p d x ) 1 p and the H 1 -norm as ‖ u ‖ : = ( ∫ ℝ N | ∇ u | 2 d x + ∫ ℝ N | u | 2 d x ) 1 2 . The symbols ⇀ and → represent weak and strong convergence in the corresponding function spaces, respectively. The notation : = and = : is used to indicate definitions, and C , C 1 , C 2 , K 1 , K 2 , ⋯ denote positive constants.

The structure of the paper is as follows: Section 2 introduces preliminary results. In Section 3, we prove the existence of mountain pass solutions, specifically addressing Theorem 1.1 (i). Section 4 explores the connection between the functional’s structure and the Pohoaev manifold, concluding the proof of Theorem 1.1 (ii).

2. Preliminarie Results

In this segment, we introduce foundational outcomes that are slated for recurrent application in the subsequent sections of the manuscript. Initially, we summarize some key inequalities.

Lemma 2.1. ([25]Gagliardo-Nirenberg inequality): For any v ∈ H 1 ( ℝ N ) , there exists a constant C N , p > 0 such that

| v | p ≤ C N , p | ∇ v | 2 γ p | v | 2 1 − γ p , γ p = N ( p − 2 ) 2 p

By Lemma 2.1, we get

∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ≤ | u 1 | r q 1 | u 2 | r q 2 ≤ C N , r m 1 ( 1 − γ q ) q 1 2 m 2 ( 1 − γ q ) q 2 2 | ∇ u 1 | 2 γ q q 1 | ∇ u 2 | 2 γ q q 2 ≤ C ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) γ q q 2 , q : = q 1 + q 2

We shall require certain findings about the Kirchhoff equation in the following proof:

(2.1) { − ( a + b ∫ ℝ N | ∇ u | 2 d x ) Δ u + V u = λ u + ν | u | p − 2 u in   ℝ N , ∫ ℝ N | u | 2 d x = m in   ℝ N ,

where N = 2 , 3 and a , b > 0 , 2 < p < 2 * . We define the energy functional of Equation (2.1) as

E ν ( u ) = a 2 | ∇ u | 2 2 + b 4 | ∇ u | 2 4 + 1 2 ∫ ℝ N V u 2 d x − ν p | u | p p .

The corresponding minimization energy is

(2.2) l ( m , ν ) : = inf u ∈ S ( m ) E ν ( u ) ,

Lemma 2.2. [26]-[28]If 2 < p < 2 + 4 N and m , ν > 0 . Subsequently,Equation(2.1) possesses an exclusive positive radial solution u 0 (up to translations)for certain values of λ 0 < 0 , and E ν ( u 0 ) = l ( m , ν ) < 0 .

Lemma 2.3.Assume V i , W i are defined in(V₁), (V₂)and(V₃). If { ( u 1 n , u 2 n ) } is bounded in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) , then we have

(i) ∫ ℝ N V i ( u i n ) 2 d x → ∫ ℝ N V i u i 2 d x , ( i = 1 , 2 ) ,

(ii) ∫ ℝ N W i ( u i n ) 2 d x → ∫ ℝ N W i u i 2 d x , ( i = 1 , 2 ) .

Proof. Since { ( u 1 n , u 2 n ) } is bounded in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) , we may assume that

( u 1 n , u 2 n ) ⇀ ( u 1 , u 2 )     in   H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) , ( u 1 n , u 2 n ) → ( u 1 , u 2 )     in   L s ( ℝ N ) × L loc s ( ℝ N )     for   s ∈ [ 1 , 2 * ) .

By lim | x | → ∞ V i ( x ) = 0 , there exists two constant M > 0 and R > 0 such that | V i | < M , ∫ B R [ ( u i n ) 2 − ( u i ) 2 ] d x < ε M and | V i | < ε M where | x | > R , then

∫ ℝ N V i [ ( u i n ) 2 − ( u i ) 2 ] d x = ∫ B R V i [ ( u i n ) 2 − ( u i ) 2 ] d x + ∫ B R c V i [ ( u i n ) 2 − ( u i ) 2 ] d x ≤ M ∫ B R [ ( u i n ) 2 − ( u i ) 2 ] d x + ε M ∫ B R c [ ( u i n ) 2 − ( u i ) 2 ] d x ≤ 2 ε → 0 ,

where n is large enough. Similarly, the result of ∫ ℝ N W i ( u i n ) 2 d x → ∫ ℝ N W i u i 2 d x can be obtained by the above proof.

Lemma 2.4. [[29], Lemma2.4] Assume that q 1 , q 2 > 1 , 2 < q 1 + q 2 < 2 * . If

( u 1 n , u 2 n ) ⇀ ( u 1 , u 2 ) in H 1 ( ℝ N ) × H 1 ( ℝ N ) ,

then up to a subsequence

∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x + ∫ ℝ N | u 1 n − u 1 | q 1 | u 2 n − u 2 | q 2 d x = ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x + o ( 1 ) .

Lemma 2.5.Suppose that the condition(M₁)is satisfied and τ > 0 , then inf ℬ ( τ ) I ( u 1 , u 2 ) < 0 .

Proof. Set u t ( x ) = t N 2 u ( t x ) , through simple calculations, we obtain

∫ ℝ N ( u t ) 2 d x = ∫ ℝ N u 2 d x , ∫ ℝ N ( ∇ u t ) 2 d x = t 2 ∫ ℝ N ( ∇ u ) 2 d x , ∫ ℝ N ( u t ) p d x = t γ p p ∫ ℝ N u p d x ,

where p ∈ ( 2 , 2 * ) . Assume that t is sufficiently small, then ( u 1 t , u 2 t ) ∈ ℬ ( τ ) , where ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) . Since

I ( u 1 t , u 2 t ) = a 2 t 2 ∑ i = 1 2 | ∇ u i | 2 2 + b 4 t 4 ∑ i = 1 2 | ∇ u i | 2 4 + 1 2 ∑ i = 1 2   ∫ ℝ N V i ( t − 1 x ) u i 2 d x     − ∑ i = 1 2 ν i p i t γ p i p i | u i | p i p i − α t γ q q ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ,

it can be readily verified that I ( u 1 t , u 2 t ) < 0 when t is sufficiently small if (M₁) holds.

Lemma 2.6.Suppose the condition(M₁)is satisfied, then there exist τ 0 = τ 0 ( m 1 , m 2 ) > 0 , α 0 = α 0 ( m 1 , m 2 ) > 0 such that for any 0 < α ≤ α 0 ,

inf ℬ ( 2 τ 0 ) \ ℬ ( τ 0 ) I ( u 1 , u 2 ) > 0.

Furthermore, there exists ε 0 > 0 , ensuring that

c ( m 1 , m 2 ) < inf ℬ ( τ 0 ) \ ℬ ( τ 0 − ε 0 ) I ( u 1 , u 2 ) .

Proof. Set τ = | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 , thus, for all ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) , we have

I ( u 1 , u 2 ) = a 2 τ + b 4 ∑ i = 1 2 | ∇ u i | 2 4 + 1 2 ∑ i = 1 2   ∫ ℝ N V i u i 2 d x − ∑ i = 1 2 ν i p i | u i | p i p i − α ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ≥ a − max ( ω 1 , ω 2 ) 2 τ + b 8 τ 2 − ∑ i = 1 2   K i | ∇ u i | 2 γ p i p i − α K 3 τ γ q q 2 ≥ b 8 τ 2 − ∑ i = 1 2   K i τ γ p i p i 2 − α K 3 τ γ q q 2 : = f ( τ ) ,

where a > a * , K i = ν i p i C N , p i m i ( 1 − γ p i ) p i 2 ( i = 1 , 2 ) and K 3 = C N , r m 1 ( 1 − γ q ) q 1 2 m 2 ( 1 − γ q ) q 2 2 . According to (M₁), we know that 0 < γ p i p i < 2 , 4 < γ q q < 2 * . Select a sufficiently large value for τ 0 > 0 ensuring that

∑ i = 1 2   K i ( τ 0 ) γ p i p i − 4 2 ≤ b 32 .

Moreover, we can take α 0 > 0 sufficiently small ensuring that

α 0 K 3 ( 2 τ 0 ) γ q q − 4 2 ≤ b 32 .

Hence, for any 0 < α ≤ α 0 and ( u 1 , u 2 ) ∈ ℬ ( 2 τ 0 ) \ ℬ ( τ 0 ) , i.e., τ 0 ≤ τ < 2 τ 0 , we have

I ( u 1 , u 2 ) ≥ b 8 τ 2 − ∑ i = 1 2   K i τ γ p i p i 2 − α K 3 τ γ q q 2 = τ 2 ( b 8 − ∑ i = 1 2   K i τ γ p i p i − 4 2 − α K 3 τ γ q q − 4 2 ) ≥ b τ 0 2 ( 1 8 − 1 32 − 1 32 ) = b 16 τ 0 2 .

According to the continuity of f ( τ ) and the fact that f ( τ 0 ) > 0 , we can find an extremely small number ε 0 > 0 such that f ( τ ) ≥ 0 when τ ∈ [ τ 0 − ε 0 , τ 0 ] . Thus

I ( u 1 , u 2 ) ≥ f ( τ ) ≥ 0 > c ( m 1 , m 2 )

for any ( u 1 , u 2 ) ∈ ℬ ( τ 0 ) ¯ \ ℬ ( τ 0 − ε 0 ) .

Lemma 2.7.Assume that 2 < p 1 , p 2 , q 1 + q 2 < 2 * . Suppose that the ( P S ) sequence { ( u 1 n , u 2 n ) } restricted on S r ( m 1 ) × S r ( m 2 ) is bounded, then we can finda ( u 1 , u 2 ) ∈ H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) and a sequence { ( λ 1 n , λ 2 n ) } ⊂ ℝ 2 such thatup to a sub-sequence

(i) ( u 1 n , u 2 n ) ⇀ ( u 1 , u 2 ) in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) , ( u 1 n , u 2 n ) → ( u 1 , u 2 ) , in L p ( ℝ N ) × L p ( ℝ N ) for p ∈ ( 2 , 2 * ) .

(ii) ( λ 1 n , λ 2 n ) → ( λ 1 , λ 2 ) in ℝ 2 .

(iii) I ′ ( u 1 n , u 2 n ) − λ 1 n ( u 1 n , 0 ) − λ 2 n ( 0 , u 2 n ) → 0 in H rad − 1 ( ℝ N ) × H rad − 1 ( ℝ N ) .

(iv) problem (1.1) admits a solution ( u 1 , u 2 ) for some λ 1 , λ 2 ≤ 0 if ( u 1 , u 2 ) satisfies the additional property J ( u 1 n , u 2 n ) → 0 , where ( λ 1 , λ 2 ) is defined by (ii).

Proof. (i) is clear. By the fact of ( I | S r ( c 1 ) × S r ( c 2 ) ) ′ ( u 1 n , u 2 n ) → 0 and Proposition 5.12 in [30], we can take two sequences of real numbers { λ 1 n } , { λ 2 n } so that

(2.3) a ∑ i = 1 2   ∫ ℝ N ∇ u i n ∇ φ i d x + b ∑ i = 1 2   ∫ ℝ N | ∇ u i n | 2 d x ∫ ℝ N ∇ u i n ∇ φ i d x + ∑ i = 1 2   ∫ ℝ N V i u i n φ i d x −   ∑ i = 1 2   ν i ∫ ℝ N | u i n | p i − 2 u i n φ i d x − α q 1 ∫ ℝ N | u 1 n | q 1 − 2 u 1 n | u 2 n | q 2 φ 1 d x −   α q 2 ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 − 2 u 2 n φ 2 d x − ∑ i = 1 2   ∫ ℝ N λ i n u i n φ i d x = o ( 1 ) ‖ ( φ 1 , φ 2 ) ‖ ,

where o ( 1 ) → 0 as n → + ∞ . For further information, it is advisable to consult [[23], Lemma 3.2]. Testing Equation (2.3) with ( u 1 n , 0 ) and ( 0 , u 2 n ) , we get

a | ∇ u 1 n | 2 2 + b | ∇ u 1 n | 2 4 + ∫ ℝ N V 1 ( u 1 n ) 2 d x − ν 1 | u 1 n | p 1 p 1 − α q 1 ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x − o ( 1 ) = λ 1 n m 1 , a | ∇ u 2 n | 2 2 + b | ∇ u 2 n | 2 4 + ∫ ℝ N V 2 ( u 2 n ) 2 d x − ν 2 | u 2 n | p 2 p 2 − α q 2 ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x − o ( 1 ) = λ 2 n m 2 .

By the boundedness of u 1 n , u 2 n in L p ( ℝ N ) and H rad 1 ( ℝ N ) for p ∈ ( 2 , 2 * ) , it can be deduced that the sequences { λ 1 n } , { λ 2 n } are bounded. Consequently, it is reasonable to presume that λ i n converges strongly to λ i ( i = 1 , 2 ) . Based on (ii) and (iii), the proof of (iv) is achievable through the method described in [[31], Proposition 2.10]. As the verification process is identical, we do not proof it here.

□

Lemma 2.8.Suppose that the Lemma 2.6 are satisfied, it follows that u 1 n converges strongly to u 1 in H rad 1 ( ℝ N ) when λ 1 < 0 .In this similar way, we can get the sequence u 2 n converges strongly to u 2 in H rad 1 ( ℝ N ) when λ 2 < 0 .

Proof. Set

(2.4) lim n → + ∞ | ∇ u i n | 2 2 = B i     ( i = 1 , 2 ) .

By Lemmas 2.3 and 2.6, we can get

∫ ℝ N V 1 ( u 1 n ) 2 d x → ∫ ℝ N V 1 u 1 2 d x , | u 1 n | p 1 p 1 → | u 1 | p 1 p 1 , ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x → ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x

and

〈 I ′ ( u 1 n , u 2 n ) − λ 1 n ( u 1 n , 0 ) , ( u 1 n , 0 ) 〉 → 0 = ( a + b B 1 ) | ∇ u 1 | 2 2 + ∫ ℝ N V 1 u 1 2 d x − ν 1 | u 1 | p 1 p 1 − α r ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x − λ 1 | u 1 | 2 2 .

Hence

a | ∇ u 1 n | 2 2 + b | ∇ u 1 n | 2 4 − λ 1 n | u 1 n | 2 2 → a | ∇ u 1 | 2 2 + b B 1 | ∇ u 1 | 2 2 − λ 1 | u 1 | 2 2 .

Since

| ∇ u 1 | 2 2 ≤ lim n → + ∞ | ∇ u 1 n | 2 2 ,     | u 1 | 2 2 ≤ lim n → + ∞ | u 1 n | 2 2 ,

we get | ∇ u 1 n | 2 2 → | ∇ u 1 | 2 2 , | u 1 n | 2 2 → | u 1 | 2 2 . □

3. Proof of Theorem 1.1 (i)

Lemma 3.1.Suppose that(M₁)is satisfied, then for α ∈ ( 0 , α 0 ) , we can take a ( P S ) sequence { ( u 1 n , u 2 n ) } for I | S r ( m 1 ) × S r ( m 2 ) at the level γ ( m 1 , m 2 ) , which satisfies ( u 1 n ) − → 0 , ( u 2 n ) − → 0 and J ( u 1 n , u 2 n ) → 0 .

Proof. Our proof approach will adhere to the methodological framework delineated in [[23], Lemma 5.5]. Let I ˜ : ℝ × ( H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) ) → ℝ defined by

I ˜ ( θ , ( u 1 , u 2 ) ) = I ( θ ∗ u 1 , θ ∗ u 2 ) ,

where θ ∗ u = e N 2 θ u ( e θ x ) . Set θ ∗ ( u 1 , u 2 ) : = ( θ ∗ u 1 , θ ∗ u 2 ) . Thus, θ ∗ ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) when ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) . We set

Γ ˜ : = { η ˜ ∈ ( S r ( m 1 ) × S r ( m 2 ) , C [ 0 , 1 ] ) : η ˜ ( 1 ) = ( 0 , η ( 1 ) ) , η ˜ ( 0 ) ∈ ( 0 , η ( 0 ) ) ,                 η ( 1 ) ∉ ℬ ( τ 0 ) ¯ , I ( η ( 1 ) ) < 0 , η ( 0 ) ∈ ℬ ( τ ¯ ) }

and

γ ˜ ( m 1 , m 2 ) : = inf η ˜ ∈ Γ ˜ max t ∈ [ 0 , 1 ] I ˜ ( η ˜ ( t ) ) .

Pay attention to the fact that γ ˜ ( m 1 , m 2 ) = γ ( m 1 , m 2 ) . Indeed, by the definitions of γ ˜ ( m 1 , m 2 ) and γ ( m 1 , m 2 ) , this equation is directly inferred from the premise that the mapps

ψ : Γ → Γ ˜ ,     η ↦ ψ ( η ) : = ( 0 , η )

and

ζ : Γ ˜ → Γ , η ˜ = ( ϑ , η ) ↦ ζ ( η ˜ ) : = ϑ ∗ η     with     ( ϑ ∗ η ) ( t ) = ϑ ( t ) ∗ η ( t )

satisfy

I ˜ ( ψ ( η ) ) = I ( η ) , I ( ζ ( η ˜ ) ) = I ˜ ( η ˜ ) .

It is observed that I ( | u 1 | , | u 2 | ) = I ( u 1 , u 2 ) holds when ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) . Then, we can assume the existence of a minimization sequence v 1 n ( t ) , v 2 n ( t ) ≥ 0 , where 0 ≤ t ≤ 1 . According to [[32], Theorem 4.1], we can find a ( P S ) γ ( m 1 , m 2 ) sequence { ( θ n , ( u 1 n , u 2 n ) ) } for I ˜ | ℝ × ( S r ( m 1 ) × S r ( m 2 ) ) and ‖ ( u 1 n , u 2 n ) − ( v 1 n , v 2 n ) ‖ → 0 . By ( u 1 n ) − → 0 , ( u 2 n ) − → 0 and direct calculation

I ˜ ( θ , ( u 1 , u 2 ) ) = I ˜ ( 0 , θ ∗ ( u 1 , u 2 ) ) , ( ∂ θ I ˜ ) ( θ , ( u 1 , u 2 ) ) = ( ∂ θ I ˜ ) ( 0 , θ ∗ ( u 1 , u 2 ) )

for u = ( u 1 , u 2 ) , φ = ( φ 1 , φ 2 ) , we have ( ∂ u I ˜ ) ( θ , u ) [ φ ] = ( ∂ u I ˜ ) ( 0 , θ ∗ u ) [ θ ∗ φ ] . Hence, { ( 0 , θ n ∗ ( u 1 n , u 2 n ) ) } is also a ( P S ) γ ( m 1 , m 2 ) sequence for I ˜ | ℝ × ( S r ( m 1 ) × S r ( m 2 ) ) . Let

( w 1 n , w 2 n ) : = θ n ∗ ( u 1 n , u 2 n ) ,

then { ( w 1 n , w 2 n ) } ⊂ S r ( m 1 ) × S r ( m 2 ) is a ( P S ) γ ( m 1 , m 2 ) sequence for I | S r ( m 1 ) × S r ( m 2 ) and then ( ∂ θ I ˜ ) ( 0 , ( w 1 n , w 2 n ) ) → 0 implies J ( w 1 n , w 2 n ) → 0 . □

According to Equation (2.4), we rewrite Equation (1.1) as

(3.1) { − ( a + b B 1 ) Δ u 1 + V 1 u 1 = λ 1 u 1 + ν 1 | u 1 | p 1 − 2 u 1 + α q 1 | u 1 | q 1 − 2 u 1 | u 2 | q 2 , − ( a + b B 2 ) Δ u 2 + V 2 u 2 = λ 2 u 2 + ν 2 | u 2 | p 2 − 2 u 2 + α q 2 | u 1 | q 1 | u 2 | q 2 − 2 u 2 .

Its corresponding Pohoaev identity as following:

(3.2) J B ( u 1 , u 2 ) : = a ∑ i = 1 2 | ∇ u i | 2 2 + b ∑ i = 1 2   B i | ∇ u i | 2 2 − 1 2 ∑ i = 1 2   ∫ ℝ N W i u i 2 d x       − ∑ i = 1 2   ν i γ p i | u i | p i p i − α γ q 1 + q 2 ( q 1 + q 2 ) ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x .

Lemma 3.2.Suppose the condition(M₁)is satisfied and 0 < α ≤ α 0 , then we can find a positive radial solution ( u 1 , u 2 ) to the system (1.1) for some ( λ 1 , λ 2 ) and I ( u 1 , u 2 ) = γ ( m 1 , m 2 ) .

Proof. By Lemma 3.1, it is possible to find a Palais-Smale sequence { ( u 1 n , u 2 n ) } for I | S r ( m 1 ) × S r ( m 2 ) at the level γ ( m 1 , m 2 ) . We first prove that { ( u 1 n , u 2 n ) } is bounded in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) . Since J ( u 1 n , u 2 n ) → 0 , we have

1 γ q q ( a ∑ i = 1 2 | ∇ u i n | 2 2 + b ∑ i = 1 2 | ∇ u i n | 2 4 − ∑ i = 1 2   ∫ ℝ N W i ( u i n ) 2 d x − ∑ i = 1 2   ν i γ p i | u i n | p i p i ) = α ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x + o ( 1 ) . γ ( m 1 , m 2 ) + o ( 1 ) = a 2 ∑ i = 1 2 | ∇ u i n | 2 2 + b 4 ∑ i = 1 2 | ∇ u i n | 2 4 + 1 2 ∑ i = 1 2   ∫ ℝ N V i ( u i n ) 2 d x − ∑ i = 1 2 ν i p i | u i n | p i p i − α ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x = a ( 1 2 − 1 γ q q ) ∑ i = 1 2 | ∇ u i n | 2 2 + b ( 1 4 − 1 γ q q ) ∑ i = 1 2 | ∇ u i n | 2 4 + 1 2 ∑ i = 1 2   ∫ ℝ N V i ( u i n ) 2 d x         − 1 γ q q ∑ i = 1 2   ∫ ℝ N W i ( u i n ) 2 d x − ∑ i = 1 2   ν i γ p i ( 1 γ p i p i − 1 γ q q ) | u i n | p i p i ≥ C ˜ ∑ i = 1 2 | ∇ u i n | 2 2 + b ( 1 4 − 1 γ q q ) ∑ i = 1 2 | ∇ u i n | 2 4 − ∑ i = 1 2   K i ν i γ p i ( 1 γ p i p i − 1 γ q q ) | ∇ u i n | 2 γ p i p i ≥ C ˜ ( | ∇ u 1 n | 2 2 + | ∇ u 2 n | 2 2 ) + b 2 ( 1 4 − 1 γ q q ) ( | ∇ u 1 n | 2 2 + | ∇ u 2 n | 2 2 ) 2         − ∑ i = 1 2   K i ν i γ p i ( 1 γ p i p i − 1 γ q q ) ( | ∇ u 1 n | 2 2 + | ∇ u 2 n | 2 2 ) γ p i p i 2 ,

where C ˜ : = ( a 2 − a γ q q − max { ω 1 , ω 2 } 2 − max { ρ 1 , ρ 2 } γ q q ) , K i = C N , p i c i ( 1 − γ p i ) p i 2 , 4 < γ q q < 2 * and 0 < γ p i p i < 2 . Therefore, { ( u 1 n , u 2 n ) } is bounded in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) Consequently, it can be inferred that

( u 1 n , u 2 n ) ⇀ ( u 1 , u 2 )   in   H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) , ( u 1 n , u 2 n ) → ( u 1 , u 2 )     in   L s ( ℝ N ) × L s ( ℝ N )     for   s ∈ ( 2 , 2 * ) .

According to Lemma 2.7, we can take a sequence { ( λ 1 n , λ 2 n ) } ⊂ ℝ 2 such that ( λ 1 n , λ 2 n ) → ( λ 1 , λ 2 ) in ℝ 2 , ( u 1 , u 2 ) is the solution of the system (3.1) and J B ( u 1 , u 2 ) = 0 .Since ( u 1 n ) − → 0 , ( u 2 n ) − → 0 , then u 1 , u 2 ≥ 0 .

Now, we prove I ( u 1 , u 2 ) = γ ( m 1 , m 2 ) . The condition J B ( u 1 n , u 2 n ) → 0 implies

(3.3) a ∑ i = 1 2 | ∇ u i n | 2 2 + b ∑ i = 1 2 | ∇ u i n | 2 4 = ∑ i = 1 2   ν i γ p i | u i n | p i p i + α γ q q ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x + ∑ i = 1 2   ∫ ℝ N W i ( u i n ) 2 d x + o ( 1 ) .

By Equation (2.4), we have

a ∑ i = 1 2 | ∇ u i n | 2 2 + b ∑ i = 1 2   B i | ∇ u i n | 2 2 = ∑ i = 1 2   ν i γ p i | u i n | p i p i + α γ q q ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x + ∑ i = 1 2   ∫ ℝ N W i ( u i n ) 2 d x + o ( 1 )

As the sequence ( u 1 n , u 2 n ) converges to ( u 1 , u 2 ) in L p ( ℝ N ) × L p ( ℝ N ) for p ∈ ( 2 , 2 * ) , then it can be inferred that ∑ i = 1 2 ν i γ p i | u i n | p i p i + α γ q q ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x + ∑ i = 1 2 ∫ ℝ N W i ( u i n ) 2 d x + o ( 1 ) of Equation (3.3) converges to

∑ i = 1 2   ν i γ p i | u i | p i p i + α γ q q ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x + ∑ i = 1 2   ∫ ℝ N W i ( u i ) 2 d x .

Combining J B ( u 1 , u 2 ) = 0 , we have

lim n → + ∞ a ∑ i = 1 2 | ∇ u i n | 2 2 + b ∑ i = 1 2   B i | ∇ u i n | 2 2 = a ∑ i = 1 2 | ∇ u i | 2 2 + b ∑ i = 1 2   B i 2 .

Hence, I ( u 1 n , u 2 n ) → I ( u 1 , u 2 ) , and then, I ( u 1 , u 2 ) = γ ( m 1 , m 2 ) . □

Proof of Theorem1.1.By Lemma 3.2, we only need to show ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) . Since ( u 1 , u 2 ) satisfies Equation (1.1), it follows that

λ 1 | u 1 | 2 2 + λ 2 | u 2 | 2 2 = a ∑ i = 1 2 | ∇ u i | 2 2 + b ∑ i = 1 2 | ∇ u i | 2 4 + ∑ i = 1 2   ∫ ℝ N V i u i 2 d x     − ∑ i = 1 2   ν i | u i | p i p i − α r ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x .

Let F ( u 1 , u 2 ) = max { ∑ i = 1 2 | u i | p i p i , ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x } , since

∑ i = 1 2 ( a − ρ i ) | ∇ u i | 2 2 ≤ a ∑ i = 1 2 | ∇ u i | 2 2 − ∑ i = 1 2   ∫ ℝ N W i u i 2 d x ≤ a ∑ i = 1 2 | ∇ u i | 2 2 + b ∑ i = 1 2 | ∇ u i | 2 4 − ∑ i = 1 2   ∫ ℝ N W i u i 2 d x ,

combining J ( u 1 , u 2 ) = 0 , we have

∑ i = 1 2 | ∇ u i | 2 2 ≤ 1 ( a − max { ρ 1 , ρ 2 } ) ( ∑ i = 1 2   ν i γ p i | u i | p i p i + α γ q q ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ) ≤ 2 ( ∑ i = 1 2   ν i γ p i + α γ q q ) F ( u 1 , u 2 ) ( a − max { ρ 1 , ρ 2 } )

and then,

λ 1 | u 1 | 2 2 + λ 2 | u 2 | 2 2 = ∑ i = 1 2   ∫ ℝ N V i u i 2 d x + ∑ i = 1 2   ∫ ℝ N W i u i 2 d x + ∑ i = 1 2 ν i ( γ p i − 1 ) | u i | p i p i       + α r ( γ q − 1 ) ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ≤ ∑ i = 1 2   ρ i | ∇ u i | 2 2 + ∑ i = 1 2 ν i ( γ p i − 1 ) | u i | p i p i + α r ( γ q − 1 ) ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ≤ C 2 F ( u 1 , u 2 ) + α r ( γ q − 1 ) ∑ i = 1 2   ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x < 0 , 2 < p i , r < 2 *

Consequently, it follows that at least one of the λ 1 or λ 2 is negative. For the sake of argument, let us assume λ 1 < 0 . According to Lemma 2.8, we have u 1 ∈ S r ( m 1 ) and u 1 n → u 1 in H rad 1 ( ℝ N ) . If λ 2 ≥ 0 , we have

− ( a + b ∫ ℝ N | ∇ u 2 | 2 d x ) Δ u 2 = λ 2 u 2 + ν 2 | u 2 | p 2 − 2 u 2 + α q 2 | u 1 | q 1 | u 2 | q 2 − 2 u 2 − V 2 u 2 ≥ 0.

From [[24], lemma A.2], we can know that u 2 = 0. Therefore, I ( u 1 , u 2 ) = I ( u 1 , 0 ) and u 1 ∈ S r ( m 1 ) is the solution of the following equation

(3.4) − ( a + b ∫ ℝ N | ∇ u 1 | 2 d x ) Δ u 1 = λ 1 u 1 + ν 1 | u 1 | p 1 − 1 u 1 − V 1 u 1 .

By lemma 2.2, u 1 is unique and I ( u 1 , 0 ) = l ( m 1 , ν 1 ) < 0 , which contradicts I ( u 1 , 0 ) = γ ( m 1 , m 2 ) > 0 . Consequently, it follows that λ 2 is negative, which implies that u 2 ∈ S r ( m 2 ) . In the end, using the maximum principle, it can be infer that u 1 , u 2 > 0 in ℝ N . □

4. Proof of Theorem 1.1(ii)

Let τ = | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 , then for each ( u 1 , u 2 ) ∈ S ( m 1 ) × S ( m 2 )

(4.1) I ( u 1 , u 2 ) = a 2 ∑ i = 1 2 | ∇ u i | 2 2 + b 4 ∑ i = 1 2 | ∇ u i | 2 4 + 1 2 ∑ i = 1 2   ∫ ℝ N V i u i 2 d x − ∑ i = 1 2 ν i p i | u i | p i p i     − α ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ≥ a 2 τ + b 4 τ 2 − max { ω 1 , ω 2 } τ − ∑ i = 1 2 ν i p i C N , p i c i ( 1 − γ p i ) p i 2 | ∇ u i | 2 γ p i p i − α K 3 τ γ q q 2 ≥ K 0 τ 2 − K 1 τ γ p 1 p 1 2 − K 2 τ γ p 2 p 2 2 − α K 3 τ γ q q 2 = : g ( τ ) ,

where a > a * , K 0 = b 4 , K i = ν i p i C N , p i c i ( 1 − γ p i ) p i 2 ( i = 1 , 2 ) , K 3 = C N , r m 1 ( 1 − γ q ) q 1 2 m 2 ( 1 − γ q ) q 2 2 , 0 < γ p i p i < 2 and 4 < γ q q < 2 * . One can easily verify that lim τ → + ∞ g ( τ ) → − ∞ and lim τ → 0 + g ( τ ) → 0 − .

Lemma 4.1.Suppose the condition(M₁)is satisfied, then we can find a constant α 1 > 0 , ensuring that g ( τ ) possesses a global maximum point at the positive level and a local minimum at the negative level, which areuniqewhen 0 < α < α 1 . Furthermore, it is observed that 0 < τ 0 < τ 1 depending on α such that g ( τ 0 ) = g ( τ 1 ) = 0 and g ( τ ) > 0 if and only if τ ∈ ( τ 0 , τ 1 ) .

Proof. Without loss of generality, we may assume that p 1 ≤ p 2 . We only show the proof for p 1 = p 2 , the proof for case p 1 < p 2 is similar. Let p = p 1 = p 2 , for τ > 0 , we have

g ( τ ) = K 0 τ 2 − ( K 1 + K 2 ) τ γ p p 2 − α K 3 τ γ q q 2 = τ γ p p 2 ( K 0 τ 4 − γ p p 2 − α K 3 τ γ q q − γ p p 2 − ( K 1 + K 2 ) ) .

Let χ ( τ ) = K 0 τ 4 − γ p p 2 − α K 3 τ γ q q − γ p p 2 , then g ( τ ) > 0 if and only if χ ( τ ) > K 1 + K 2 . Clearly, χ ( τ ) possesses a maximum point which is unique

τ ¯ = ( K 0 ( 4 − γ p p ) α K 3 ( γ q q − γ p p ) ) 2 γ q q − 4

and

χ ( τ ¯ ) = C α − 4 − γ p p γ q q − 4 ,

where 0 < γ p i p i < 2 , 4 < γ q q < 2 * and C > 0 . Therefore, we are able to make α as small as necessary, ensuring that χ ( τ ¯ ) > K 1 + K 2 . Then, we can find a constant α 1 > 0 , ensuring that g ( τ ) > 0 on τ ∈ ( τ 0 , τ 1 ) when 0 < α < α 1 . Since g ( τ ) → 0 − as τ → 0 + , it is evident that g ( τ ) can find a local minimum on ( 0 , τ 0 ) . Therefore, g ( τ ) must possess a minimum of two critical points. Set h ( τ ) : = 2 K 0 τ 4 − γ p p 2 − γ q q 2 α K 3 τ γ q q − γ p p 2 , then

g ′ ( τ ) = 2 K 0 τ − γ p p 2 ( K 1 + K 2 ) τ γ p p − 2 2 − γ q q 2 α K 3 τ γ q q − 2 2 = τ γ p p − 2 2 ( h ( τ ) − γ p p 2 ( K 1 + K 2 ) ) .

So g ′ ( τ ) = 0 if and only if h ( τ ) = γ p p 2 ( K 1 + K 2 ) . As the presence of a sole global maximum for h ( τ ) , it leads to h ( τ ) = γ p p 2 ( K 1 + K 2 ) possesses no more than two solutions, i.e., g ( τ ) possesses no more than two points. □

Remark 4.1.If p 1 < p 2 , we have

g ( τ ) = K 0 τ 2 − K 1 τ γ p 1 p 1 2 − K 2 τ γ p 2 p 2 2 − α K 3 τ γ q q 2 = τ γ p 1 p 1 2 ( K 0 τ 4 − γ p 1 p 1 2 − K 2 τ γ p 2 p 2 − γ p 1 p 1 2 − α K 3 τ γ q q − γ p 1 p 1 2 − K 1 ) = : τ γ p 1 p 1 2 ( χ ( τ ) − K 1 ) .

Through computational analysis, it is easy to know that χ ( τ ) exists a global maximum point τ ¯ with

τ ¯ > ( C 1 C 2 α ) 2 γ q q − 4 = : τ ˜ ,

where C 1 = K 0 ( 4 − γ p 1 p 1 ) ( 4 − γ p 2 p 2 ) 4 , C 2 = K 3 ( γ q q − γ p 1 p 1 ) ( γ q q − γ p 2 p 2 ) 4 . Moreover,

χ ( τ ¯ ) > χ ( τ ˜ ) > τ ˜ γ p 2 p 2 − γ p 1 p 1 2 ,

when α is sufficiently small. Therefore, we are able to make α as small as necessary, ensuring that χ ( τ ¯ ) > K 1 . The case of p 1 > p 2 is similar.

Lemma 4.2.Suppose the condition(M₁)is satisfied, then we can find a constant α 2 > 0 , ensuring that ℳ 0 = ∅ when 0 < α < α 2 . The ℳ is a submanifold of C 1 with a codimension of three in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) .

Proof. Assume that there exists ( u 1 , u 2 ) ∈ ℳ 0 , then

(4.2) ( ϒ ( u 1 , u 2 ) ) ( 0 ) = a ∑ i = 1 2 | ∇ u i | 2 2 + b ∑ i = 1 2 | ∇ u i | 2 4 − ∑ i = 1 2   ∫ ℝ N W i u i 2 d x = ∑ i = 1 2   ν i γ p i | u i | p i p i + α γ q q ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ,

and

(4.3) ( ϒ ( u 1 , u 2 ) ) ( 0 ) = 2 a ∑ i = 1 2 | ∇ u i | 2 2 + 4 b ∑ i = 1 2 | ∇ u i | 2 4 + ∑ i = 1 2   ∫ ℝ N Z i u i 2 d x = ∑ i = 1 2   ν i γ p i 2 p i | u i | p i p i + α ( γ q q ) 2 ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x .

Combine (4.2) and (4.3), we have

( γ q q − 2 ) a ∑ i = 1 2 | ∇ u i | 2 2 + ( γ q q − 4 ) b ∑ i = 1 2 | ∇ u i | 2 4 = ∑ i = 1 2   ν i γ p i ( γ q q − γ p i p i ) | u i | p i p i + ∑ i = 1 2   ∫ ℝ N Y i u i 2 d x

and then

b 2 ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) 2 ≤ b ( | ∇ u 1 | 2 4 + | ∇ u 2 | 2 4 ) + a ( γ q q − 2 ) γ q q − 4 ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) = 1 γ q q − 4 ∑ i = 1 2   ν i γ p i ( γ q q − γ p i p i ) | u i | p i p i + 1 γ q q − 4 ∑ i = 1 2   ∫ ℝ N Y i u i 2 d x ≤ C 3 ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) γ p i p i 2 + C 4 ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) ,

where C 4 = max { σ 1 , σ 2 } γ q q − 4 . Hence, there exists C 5 > 0 , independent of α such that | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ≤ C 5 . By (4.2) and (4.3), we can get

∑ i = 1 2   ν i γ p i ( 2 − γ p i p i ) | u i | p i p i ≤ 2 b ∑ i = 1 2 | ∇ u i | 2 4 + ∑ i = 1 2   ν i γ p i ( 2 − γ p i p i ) | u i | p i p i = α γ q q ( γ q q − 2 ) ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x − ∑ i = 1 2   ∫ ℝ N ( 2 W i − Z i ) u i 2 d x ≤ α γ q q ( γ q q − 2 ) ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x + ∑ i = 1 2 ( 2 ρ i + σ i ) | ∇ u i | 2 2 .

Suppose ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x = 0 , we can infer that u 1 = u 2 = 0 . There is a conflict. So ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ≠ 0 , then

a ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) ≤ b ( | ∇ u 1 | 2 4 + | ∇ u 2 | 2 4 ) + a ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) = ∑ i = 1 2   ν i γ p i | u i | p i p i + α γ q q ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x + ∑ i = 1 2   ∫ ℝ N W i u i 2 d x ≤ ( γ q q − 2 2 − max { γ p 1 p 1 , γ p 2 p 2 } + 1 ) α γ q q ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x + ∑ i = 1 2   ρ i | ∇ u i | 2 2       + ( 1 2 − max { γ p 1 p 1 , γ p 2 p 2 } ) ∑ i = 1 2 ( 2 ρ i + σ i ) | ∇ u i | 2 2 ≤ α C 6 ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) γ q q 2 + C 7 ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) ,

where C 6 > 0 , C 7 = ( 2 max { ρ 1 , ρ 2 } + max { σ 1 , σ 2 } 2 − max { γ p 1 p 1 , γ p 2 p 2 } ) + max { ρ 1 , ρ 2 } and then, | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ≥ ( a − C 7 α C 6 ) 2 γ q q − 2 where a > a * . Hence

( a − C 7 α C 6 ) 2 γ q q − 2 ≤ | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ≤ C 5 .

There is unachievable when α is sufficiently small.

Next, we verify that ℳ is a submanifold of C 1 with a codimension of three in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) . Note that

ℳ : = { ( u 1 , u 2 ) ∈ H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) : J ( u 1 , u 2 ) = 0 , D 1 ( u 1 ) = 0 , D 2 ( u 2 ) = 0 } ,

where D 1 ( u 1 ) = | u 1 | 2 2 − m 1 , D 2 ( u 2 ) = | u 2 | 2 2 − m 2 . We just have to attest this map

d ( J , D 1 , D 2 ) : H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) → ℝ 3

is a surjective. Otherwise, by the independence of d D 1 ( u 1 ) and d D 2 ( u 2 ) , d J ( u 1 , u 2 ) must be a linear combination of d D 1 ( u 1 ) and d D 2 ( u 2 ) , i.e., there existn ν 1 , ν 2 ∈ ℝ such that

d J ( u 1 , u 2 ) = ν 1 d D 1 ( u 1 ) + ν 2 d D 2 ( u 2 ) ,

that is, ( u 1 , u 2 ) satisfies the following system

{ − ( a + 2 b ∫ ℝ N | ∇ u 1 | 2 d x ) Δ u 1 + W 1 u 1 = ν 1 u 1 + ν 1 γ p 1 p 1 2 | u 1 | p 1 − 2 u 1 + α q 1 γ q q 2 | u 1 | q 1 − 2 u 1 | u 2 | q 2 , − ( a + 2 b ∫ ℝ N | ∇ u 2 | 2 d x ) Δ u 2 + W 2 u 2 = ν 2 u 2 + ν 2 γ p 2 p 2 2 | u 2 | p 2 − 2 u 2 + α q 2 γ q q 2 | u 1 | q 1 | u 2 | q 2 − 2 u 2 .

According to the Pohoaev identity, we can get

2 a ∑ i = 1 2 | ∇ u i | 2 2 + 4 b ∑ i = 1 2 | ∇ u i | 2 4 + ∑ i = 1 2   ∫ ℝ N Z i u i 2 d x = ∑ i = 1 2   ν i γ p i 2 p i | u i | p i p i + α ( γ q q ) 2 ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x

and then ( u 1 , u 2 ) ∈ ℳ 0 , there is a conflict. □

Remark 4.2.We can observe that ℳ = ℳ + ∪ ℳ − , where ℳ + ∩ ℳ − = ∅ and α ∈ ( 0 , α 2 ) .

We define α 0 = min { α 1 , α 2 } , V + : = { ( u 1 , u 2 ) ∈ H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) : ∫ ℝ N | u 1 | q 1 ⋅ | u 2 | q 2 d x > 0 } and V 0 : = { ( u 1 , u 2 ) ∈ H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) : ∫ ℝ N | u 1 | q 1 | u 2 | q 2 = 0 } . Next, we begin to investigate the properties of ϒ ( u 1 , u 2 ) ( s ) .

Lemma 4.3.Suppose the condition(M₁)is satisfied. For all ( u 1 , u 2 ) ∈ ( S r ( m 1 ) × S r ( m 2 ) ) ∩ V + , we can infer that the function ϒ ( u 1 , u 2 ) ( θ ) possesses two zeros s ( u 1 , u 2 ) < t ( u 1 , u 2 ) and two critical points c ( u 1 , u 2 ) < d ( u 1 , u 2 ) satisfying the inequality c ( u 1 , u 2 ) < s ( u 1 , u 2 ) < d ( u 1 , u 2 ) < t ( u 1 , u 2 ) , where 0 < α < α 0 . Additionally

(i) θ ∗ ( u 1 , u 2 ) ∈ ℳ + ⇔ θ = c ( u 1 , u 2 ) and θ ∗ ( u 1 , u 2 ) ∈ ℳ − ⇔ θ = d ( u 1 , u 2 ) .

(ii) | ∇ ( θ ∗ ( u 1 , u 2 ) ) | 2 2 < τ 0 for every θ < s ( u 1 , u 2 ) and

I ( c ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ) = min { I ( θ ∗ ( u 1 , u 2 ) ) : θ ∈ ℝ , | ∇ ( θ ∗ ( u 1 , u 2 ) ) | 2 2 < τ 0 } < 0

(iii) I ( d ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ) = max s ∈ ℝ I ( θ ∗ ( u 1 , u 2 ) ) .

(iv) The maps ( u 1 , u 2 ) ↦ c ( u 1 , u 2 ) and ( u 1 , u 2 ) ↦ d ( u 1 , u 2 ) belongs to C 1 .

Proof. Clearly, we can get

( ϒ ( u 1 , u 2 ) ) ( θ ) = a e 2 θ ∑ i = 1 2 | ∇ u i | 2 2 + b e 4 θ ∑ i = 1 2 | ∇ u i | 2 4 − ∑ i = 1 2 ∫ ℝ N W i ( e − θ x ) u i 2 d x     − ∑ i = 1 2   ν i γ p i e γ p i p i θ | u i | p i p i − α γ q q e γ q q θ ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x = a ∑ i = 1 2 | ∇ ( θ ∗ u i ) | 2 2 + b ∑ i = 1 2 | ∇ ( θ ∗ u i ) | 2 4 − ∑ i = 1 2 ∫ ℝ N W i ( θ ∗ u i ) 2 d x     − ∑ i = 1 2   ν i γ p i | θ ∗ u i | p i p i − α γ q q ∫ ℝ N | θ ∗ u 1 | q 1 | θ ∗ u 2 | q 2 d x = J ( θ ∗ u 1 , θ ∗ u 2 ) ,     ∀ ( u 1 , u 2 ) ∈ ( S r ( m 1 ) × S r ( m 2 ) ) ∩ V + . (4.1) θ ∗ ( u 1 , u 2 ) ∈ ℳ ⇔ ( ϒ ( u 1 , u 2 ) ) ′ ( θ ) = 0 ϒ ( u 1 , u 2 ) ( θ ) = I ( θ ∗ u 1 , θ ∗ u 2 ) ≥ g ( e 2 θ τ ) ,

where τ = ∑ i = 1 2 | ∇ u i | 2 2 . Hence, if e 2 θ τ ∈ ( τ 0 , τ 1 ) ,i.e., θ ∈ ( 1 2 ln τ 0 τ , 1 2 ln τ 1 τ ) , we get ϒ ( u 1 , u 2 ) ( θ ) > 0 . By ϒ ( u 1 , u 2 ) ( − ∞ ) = 0 − and ϒ ( u 1 , u 2 ) ( + ∞ ) = − ∞ , we can infer that ϒ ( u 1 , u 2 ) ( θ ) possesses two critical points. Here c ( u 1 , u 2 ) represents the local minimum, d ( u 1 , u 2 ) denotes the global maximum and c ( u 1 , u 2 ) < d ( u 1 , u 2 ) . In addition, they satisfy the following

c ( u 1 , u 2 ) < 1 2 ln τ 0 τ < d ( u 1 , u 2 ) < 1 2 ln τ 1 τ .

As with Lemma 4.1, we can prove that ϒ ( u 1 , u 2 ) ( θ ) possesses no more than two critical points. Hence, ϒ ( u 1 , u 2 ) ( θ ) possesses precisely two points of criticality.

Observe the fact that θ ∗ ( u 1 , u 2 ) ∈ ℳ ⇔ ( ϒ ( u 1 , u 2 ) ) ( θ ) = 0 , then θ ∗ ( u 1 , u 2 ) ∈ ℳ ⇔ θ ∈ { c ( u 1 , u 2 ) , d ( u 1 , u 2 ) } . As we know that c ( u 1 , u 2 ) is a local minimum point, so

( ϒ c ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ) ′ ′ ( 0 ) = ( ϒ ( u 1 , u 2 ) ) ′ ′ ( c ( u 1 , u 2 ) ) ≥ 0.

Since ℳ 0 = ∅ , so ( ϒ c ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ) ′ ′ ( 0 ) > 0 , and then c ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ∈ ℳ + . Similarly, d ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ∈ ℳ ′ − . Furthermore, due to the monotonic nature and considering the asymptotic behavior, ϒ ( u 1 , u 2 ) ( θ ) possesses precisely two zeros c ( u 1 , u 2 ) < d ( u 1 , u 2 ) and they satisfy c ( u 1 , u 2 ) < s ( u 1 , u 2 ) < d ( u 1 , u 2 ) < t ( u 1 , u 2 ) .

It remains to show that the maps ( u 1 , u 2 ) ↦ c ( u 1 , u 2 ) and ( u 1 , u 2 ) ↦ d ( u 1 , u 2 ) are of class C 1 . Utilizing the theorem of implicit function on T ( θ , u 1 , u 2 ) : = ( ϒ ( u 1 , u 2 ) ) ′ ( θ ) , and using facts that

T ( c ( u 1 , u 2 ) , u 1 , u 2 ) = T ( d ( u 1 , u 2 ) , u 1 , u 2 ) = 0 , ∂ θ T ( c ( u 1 , u 2 ) , u 1 , u 2 ) = ( ϒ ( u 1 , u 2 ) ) ( c ( u 1 , u 2 ) ) > 0 , ∂ θ T ( d ( u 1 , u 2 ) , u 1 , u 2 ) = ( ϒ ( u 1 , u 2 ) ) ′ ′ ( d ( u 1 , u 2 ) ) < 0

and the reality that a continuous transition from ℳ + to ℳ − is impossible. Thus, the analysis shows that ( u 1 , u 2 ) ↦ c ( u 1 , u 2 ) and ( u 1 , u 2 ) ↦ d ( u 1 , u 2 ) belong to the C 1 class. □

Lemma 4.4.Suppose the condition(M₁)is satisfied. For all ( u 1 , u 2 ) ∈ ( S r ( m 1 ) × S r ( m 2 ) ) ∩ V 0 , we have ϒ ( u 1 , u 2 ) ( s ( u 1 , u 2 ) ) = 0 and ( ϒ ( u 1 , u 2 ) ) ′ ( c ( u 1 , u 2 ) ) = 0 such that s ( u 1 , u 2 ) < c ( u 1 , u 2 ) . Additionally

(i) ℳ = ℳ + and θ ∗ ( u 1 , u 2 ) ∈ ℳ + ⇔ θ = s ( u 1 , u 2 ) .

(ii) I ( c ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ) = min θ ∈ ℝ I ( θ ∗ ( u 1 , u 2 ) ) .

(iii) | ∇ ( θ ∗ u 1 ) | 2 2 + | ∇ ( θ ∗ u 2 ) | 2 2 < τ 0 where θ < s ( u 1 , u 2 ) .

Proof. We can suppose that p 1 ≤ p 2 which is without generality loss. Clearly, ϒ ( u 1 , u 2 ) ( θ ) → + ∞ as θ → + ∞ and ϒ ( u 1 , u 2 ) ( θ ) → 0 − as θ → − ∞ where

( u 1 , u 2 ) ∈ ( S r ( m 1 ) × S r ( m 2 ) ) ∩ V 0 . Hence, the function ϒ ( u 1 , u 2 ) ( θ ) attains its

global minimum at the point c ( u 1 , u 2 ) , which is below the zero level. In order to prove the critical point of ϒ ( u 1 , u 2 ) ( θ ) is unique, we can see that ( ϒ ( u 1 , u 2 ) ) ′ ( θ ) = 0 is equivalent to

a e ( 2 − γ p 1 p 1 ) θ ∑ i = 1 2 | ∇ u i | 2 2 + b e ( 4 − γ p 1 p 1 ) θ ∑ i = 1 2 | ∇ u i | 2 4 −   e ( − γ p 1 p 1 ) θ ∑ i = 1 2   ∫ ℝ N W i ( e − θ x ) u i 2 d x − ν 2 γ p 2 e ( γ p 2 p 2 − γ p 1 p 1 ) θ | u 2 | p 2 p 2 = ν 1 γ p 1 | u 1 | p 1 p 1 .

Through some calculation analysis, it becomes evident that the equation possesses a single solution. Therefore, θ ∗ ( u 1 , u 2 ) ∈ ℳ ⇔ θ = s ( u 1 , u 2 ) . By minimality ( ϒ ( u 1 , u 2 ) ) ′ ′ ( s ( u 1 , u 2 ) ) ≥ 0 , and since ℳ 0 = ∅ , we deduce that ( ϒ c ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ) ′ ′ ( 0 ) > 0 , and then c ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ∈ ℳ + . Furthermore, due to the monotonic nature and considering the asymptotic behavior, the function ϒ ( u 1 , u 2 ) ( θ ) possesses a sole zero point s ( u 1 , u 2 ) and satisfying c ( u 1 , u 2 ) < s ( u 1 , u 2 ) . As ϒ ( u 1 , u 2 ) ( θ ) ≥ g ( e 2 θ ( | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 ) ) , then ϒ ( u 1 , u 2 ) ( θ ) ≥ g ( τ 0 ) = 0 at s = 1 2 ln τ 0 | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 . Hence, c ( u 1 , u 2 ) ≤ 1 2 ln τ 0 | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 . It follows that | ∇ ( θ ∗ u 1 ) | 2 2 + | ∇ ( θ ∗ u 2 ) | 2 2 < τ 0 where θ < c ( u 1 , u 2 ) . □

Remark 4.3.By Lemma4.3 and4.4, we can get θ ∗ ( u 1 , u 2 ) ∈ ℳ + ⇔ θ = s ( u 1 , u 2 ) and s ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ∈ ℬ ( τ 0 ) where ( u 1 , u 2 ) ∈ S r ( m 1 ) × S r ( m 2 ) .

According to Equation (1.11), we have c ( m 1 , m 2 ) = inf ℬ ( τ 0 ) I ( u 1 , u 2 ) < 0 , we will study the properties of c ( m 1 , m 2 ) as follows.

Lemma 4.5.Suppose the condition(M₁)is satisfied. When 0 < α < α 0 , we have

c ( m 1 , m 2 ) = inf ℳ I ( u 1 , u 2 ) = inf ℳ + I ( u 1 , u 2 ) ,

and there is a positive ε 0 sufficiently small, ensuring that

c ( m 1 , m 2 ) < inf ℬ ( τ 0 ) ¯ \ ℬ ( τ 0 − ε 0 ) I ( u 1 , u 2 ) .

Proof. We first prove c ( m 1 , m 2 ) = inf ℳ + I ( u 1 , u 2 ) . For every ( u 1 , u 2 ) ∈ ℳ + , c ( u 1 , u 2 ) = 0 . It follows from Lemma 4.3 and Lemma 4.4 that 0 < 1 2 ln τ 0 | ∇ u 1 | 2 2 + | ∇ u 2 | 2 2 , namely, | ∇ u 1 | 2 2 + | ∇ u 1 | 2 2 < τ 0 . Thus, ℳ + ⊂ ℬ ( τ 0 ) , and then c ( m 1 , m 2 ) ≤ inf ℳ + I ( u 1 , u 2 ) . For every ( u 1 , u 2 ) ∈ ℬ ( τ 0 ) , there exists a unique c ( u 1 , u 2 ) ∈ ℝ , such that

c ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ∈ ℳ + ⊂ ℬ ( τ 0 ) .

According to Lemma 4.3 (ii) and Lemma 4.4, it can infer that

I ( c ( u 1 , u 2 ) ∗ ( u 1 , u 2 ) ) = min { I ( θ ∗ ( u 1 , u 2 ) ) : s ∈ ℝ , | ∇ ( θ ∗ ( u 1 , u 2 ) ) | 2 2 < τ 0 } ≤ I ( u 1 , u 2 ) .

Thus, inf ℳ + I ( u 1 , u 2 ) ≤ c ( m 1 , m 2 ) . To sum up, we have c ( m 1 , m 2 ) = inf ℳ + I ( u 1 , u 2 ) . By 4.3 (iii) and Lemma 4.4, we get that

inf ℳ I ( u 1 , u 2 ) = inf ℳ + I ( u 1 , u 2 ) .

Next, we have to proof c ( m 1 , m 2 ) < inf ℬ ( τ 0 ) ¯ \ ℬ ( τ 0 − ε 0 ) I ( u 1 , u 2 ) . As g ( τ 0 ) = 0 , considering the function g is continuous, there is a positive ε 0 sufficiently small, ensuring that g ( τ ) ≥ c ( m 1 , m 2 ) 2 where τ ∈ [ τ 0 − ε 0 , τ 0 ] . Therefore,

I ( u 1 , u 2 ) ≥ g ( τ ) ≥ c ( m 1 , m 2 ) 2 > c ( m 1 , m 2 )

for every ( u 1 , u 2 ) ∈ ℬ ( τ 0 ) ¯ \ ℬ ( τ 0 − ε 0 ) . □

Lemma 4.6.Suppose the condition(M₁)is satisfied. When 0 < α < α 0 , we have c ( m 1 , m 2 ) < min { l ( m 1 , ν 1 ) , l ( m 2 , ν 2 ) } .

Proof. We only prove c ( m 1 , m 2 ) < l ( m 1 , ν 1 ) . For every ( u 1 , u 2 ) ∈ ℬ ( τ 0 )

I ( u 1 , u 2 ) = E ν 1 ( u 1 ) + E ν 2 ( u 2 ) − α ∫ ℝ N | u 1 | q 1 | u 2 | q 2 d x ≤ E ν 1 ( u 1 ) + E ν 2 ( u 2 ) .

Thus, c ( m 1 , m 2 ) ≤ inf ℬ ( τ 0 ) ( E ν 1 ( u 1 ) + E ν 2 ( u 2 ) ) . For all u 1 ∈ S r ( m 1 ) satisfying | ∇ u 1 | 2 2 = τ 0 , as defined in Lemma 4.1, we have

E ν 1 ( u 1 ) = a 2 | ∇ u 1 | 2 2 + b 4 | ∇ u 1 | 2 4 + 1 2 ∫ ℝ N V 1 u 1 2 d x − ν 1 p 1 | u 1 | p 1 p 1 ≥ K 0 τ 0 2 − K 1 τ 0 γ p 1 p 1 2 ≥ g ( τ 0 ) = 0 ,

where K 0 , K 1 are given by equation (4.1). Thus

inf S r ( m 1 ) E ν 1 = inf B ( m 1 , τ 0 ) E ν 1 < 0 ,

where B ( c , τ ) : = { u ∈ S r ( c ) : | ∇ u | 2 2 < τ } . In addition, because τ ↦ K 0 τ 2 − K 1 τ γ p p 2 is continuous, by using the proof from Lemma 4.5 to show that we can find a sufficiently small constant ε 0 > 0 , ensuring that

l ( m 1 , ν 1 ) < inf B ( m 1 , τ 0 ) ¯ \ B ( m 1 , τ 0 − ε 0 ) E ν 1 ( u 1 ) ,

and one can easily confirm that inf B ( m 2 , ε 0 ) E ν 2 ( u 2 ) < 0 . Let

U = { ( u 1 , u 2 ) : u 1 ∈ B ( m 1 , τ 0 − ε 0 ) , u 2 ∈ B ( m 2 , ε 0 ) } ,

then U ⊂ ℬ ( τ 0 ) . Hence

inf ℬ ( τ 0 ) ( E ν 1 ( u 1 ) + E ν 2 ( u 2 ) ) ≤ inf U ( E ν 1 ( u 1 ) + E ν 2 ( u 2 ) ) = inf B ( m 1 , τ 0 − ε 0 ) E ν 1 ( u 1 ) + inf B ( m 2 , ε 0 ) E ν 2 ( u 2 ) < inf B ( m 1 , τ 0 − ε 0 ) E ν 1 ( u 1 ) = l ( m 1 , ν 1 ) .

Therefore, c ( m 1 , m 2 ) < l ( m 1 , ν 1 ) . Similarly, c ( m 1 , m 2 ) < l ( m 2 , ν 2 ) . □

Lemma 4.7.Take the sequence { ( u 1 n , u 2 n ) } ⊂ S r ( m 1 ) × S r ( m 2 ) as a ( P S ) c ( m 1 , m 2 ) sequence for I | S r ( m 1 ) × S r ( m 2 ) and J ( u 1 n , u 2 n ) → 0 as n → + ∞ . Then, ( u 1 n , u 2 n ) → ( u 1 , u 2 ) in H r a d 1 ( ℝ N ) × H r a d 1 ( ℝ N ) .

Proof. Because I ( | u 1 n | , | u 2 n | ) = I ( u 1 n , u 2 n ) , we might as well assume u 1 n , u 2 n ≥ 0 . To begin with, we demonstrate that the sequence { ( u 1 n , u 2 n ) } is bounded in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) 1 . As { ( u 1 n , u 2 n ) } is a ( P S ) c ( m 1 , m 2 ) sequence for I restricted on S r ( m 1 ) × S r ( m 2 ) , we conclude that

(4.4) I ( u 1 n , u 2 n ) → c ( m 1 , m 2 ) , (4.5) ‖ ( I | S r ( m 1 ) × S r ( m 2 ) ) ′ ( u 1 n , u 2 n ) ‖ ( H 1 ( ℝ N ) ) * → 0 , (4.6) J ( u 1 n , u 2 n ) → 0 , n → + ∞ .

By Equations (4.4) and (4.6), we get

c ( m 1 , m 2 ) + o ( 1 ) = a ( 1 2 − 1 γ q q ) ∑ i = 1 2 | ∇ u i n | 2 2 + b ( 1 4 − 1 γ q q ) ∑ i = 1 2 | ∇ u i n | 2 4 + 1 2 ∑ i = 1 2   ∫ ℝ N V i u i 2 d x         + 1 γ q q ∑ i = 1 2 ∫ ℝ N W i u i 2 d x ∑ i = 1 2   ν i γ p i ( 1 γ p i p i − 1 γ q q ) | u i n | p i p i ≥ ( a − max { ω 1 , ω 2 } 2 − a − min { ρ 1 , ρ 2 } γ q q ) τ + b 2 ( 1 4 − 1 γ q q ) τ 2         − ∑ i = 1 2   C N , p i γ p i ( 1 γ p i p i − 1 γ q q ) c i ( 1 − γ p i ) p i 2 ν i | ∇ u i n | 2 γ p i p i ≥ ( a − max { ω 1 , ω 2 } 2 − a − min { ρ 1 , ρ 2 } γ q q ) τ + b 2 ( 1 4 − 1 γ q q ) τ 2 − ∑ i = 1 2   C i τ γ p i p i 2 ,

where τ = ∑ i = 1 2 | ∇ u i | 2 2 , 0 < γ p i p i < 2 , 4 < γ q q < 2 * , C i > 0 ( i = 1 , 2 ) . Thus, ( u 1 n , u 2 n ) is bounded in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) , and we may assume that

( u 1 n , u 2 n ) ⇀ ( u 1 , u 2 )     in   H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) , ( u 1 n , u 2 n ) → ( u 1 , u 2 )     in   L s ( ℝ N ) × L s ( ℝ N ) , s ∈ ( 2 , 2 * ) .

According to Equation (4.5), we can find two real-valued sequences { λ 1 n } , { λ 2 n } , such that

o ( 1 ) ‖ ( φ 1 , φ 2 ) ‖ = a ∑ i = 1 2   ∫ ℝ N ∇ u i n ∇ φ i d x + b ∑ i = 1 2   ∫ ℝ N | ∇ u i n | 2 d x ∫ ℝ N ∇ u i n ∇ φ i d x     + ∑ i = 1 2   ∫ ℝ N V i n u i n φ i d x − ∑ i = 1 2   ν i ∫ ℝ N | u i n | p i − 2 u i n φ i d x     − α q 1 ∫ ℝ N | u 1 n | q 1 − 2 u 1 n | u 2 n | q 2 φ 1 d x − α q 2 ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 − 2 u 2 n φ 2 d x     − ∑ i = 1 2   ∫ ℝ N λ i n u i n φ i d x ,

where o ( 1 ) → 0 as n → + ∞ . By Lemma 2.7, λ i n → λ i for i = 1 , 2 and ( u 1 , u 2 ) is a solution to the system (1.1). Since I ( u 1 , u 2 ) ≤ lim n → ± ∞ I ( u 1 n , u 2 n ) = c ( m 1 , m 2 ) < 0 , then ( u 1 , u 2 ) ≠ ( 0 , 0 ) . According to the argument of Theorem 1.1 (i), at least one of λ 1 , λ 2 is less than zero. If λ 1 < 0 , based on the information provided in Lemma 2.6, we can deduce that u 1 n → u 1 in H rad 1 ( ℝ N ) . Subsequently, we show that λ 2 < 0 . If λ 2 is not less than zero, then

( a + b ∫ ℝ N | ∇ u 2 | 2 d x ) Δ u 2 + V 2 u 2 = λ 2 u 2 + ν 2 | u 2 | p 2 − 2 u 2 + α q 2 | u 1 | q 1 | u 2 | q 2 − 2 u 2 ≥ 0

and u 1 satisfies the following equation

{ − ( a + b ∫ ℝ N | ∇ u 1 | 2 d x ) Δ u 1 + V 1 u 1 = λ u 1 + ν | u 1 | p − 2 u 1 in   ℝ N , ∫ ℝ N | u 1 | 2 d x = m 1 in   ℝ N ,

where 2 < p 1 < 2 + 4 N . By Lemma 2.1, u 1 is unique and then, E ν 1 ( u 1 ) = l ( m 1 , ν 1 ) . Thus

c ( m 1 , m 2 ) = lim n → ∞ I ( u 1 n , u 2 n ) = lim n → ∞ a 2 ∑ i = 1 2 | ∇ u i n | 2 2 + b 4 ∑ i = 1 2 | ∇ u i n | 2 4 + ∑ i = 1 2   ∫ ℝ N V i n ( u i n ) 2 d x     − ∑ i = 1 2 ν i p i | u i n | p i p i − α ∫ ℝ N | u 1 n | q 1 | u 2 n | q 2 d x = lim n → ∞ a 2 ∑ i = 1 2 | ∇ u i n | 2 2 + b 4 ∑ i = 1 2 | ∇ u i n | 2 4 + ∑ i = 1 2   ∫ ℝ N V i n ( u i n ) 2 d x − ν 1 p 1 | u 1 n | p 1 p 1 ≥ a 2 | ∇ u 1 | 2 2 + b 4 | ∇ u 1 | 2 4 + ∫ ℝ N V 1 u 1 2 d x − ν 1 p 1 | u 1 | p 1 p 1 = l ( m 1 , ν 1 ) .

This results in a conflict with the conclusions of Lemma 4.6. Consequently, we can infer that λ 2 < 0 which implies that u 2 n → u 2 in H rad 1 ( ℝ N ) . □

Proof of Theorem1.1. (ii). Let { ( u 1 n , u 2 n ) } ⊂ ℬ ( τ 0 ) be a minimizing sequence for c ( m 1 , m 2 ) , that is

I ( u 1 n , u 2 n ) → c ( m 1 , m 2 ) .

By Lemmas 4.3 and 4.4, s ( u 1 n , u 2 n ) ∗ ( u 1 n , u 2 n ) ∈ ℳ + for every n , | ∇ ( s ( u 1 n , u 2 n ) ∗ ( u 1 n , u 2 n ) ) | 2 2 < τ 0 and

I ( s ( u 1 n , u 2 n ) ∗ ( u 1 n , u 2 n ) ) ≤ I ( u 1 n , u 2 n ) .

Let ( w 1 n , w 2 n ) : = s ( u 1 n , u 2 n ) ∗ ( u 1 n , u 2 n ) , then { ( w 1 n , w 2 n ) } ⊂ ℬ ( τ 0 ) serves as a minimizing sequence for c ( m 1 , m 2 ) and ( w 1 n , w 2 n ) ∈ ℳ + . By Lemma 4.5, { ( w 1 n , w 2 n ) } ⊂ A ( τ 0 − ε 0 ) . Thus, Ekeland’s variational principle yields the existence of a new minimizing sequence { ( v 1 n , v 2 n ) } , which is also a Palais-Smale sequence for I | S r ( m 1 ) × S r ( m 2 ) and ‖ ( w 1 n , w 2 n ) − ( v 1 n , v 2 n ) ‖ → 0 . Hence, { ( v 1 n , v 2 n ) } ⊂ ℬ ( τ 0 ) and J ( v 1 n , v 2 n ) → 0 . It follows from Lemma 4.7 that there exists v 1 , v 2 > 0 such that ( v 1 n , v 2 n ) → ( v 1 , v 2 ) in H rad 1 ( ℝ N ) × H rad 1 ( ℝ N ) , and then, I | A ( τ 0 ) attains a local minimum at ( v 1 , v 2 ) . Accordingly, ( v 1 , v 2 ) is a solution for Equations (1.1)-(1.2) for some λ 1 , λ 2 < 0 , which is positive and radially.

□

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